We begin with two (well-known) observations on the affinoid adic space \(\mathrm {Spa}(A,A^\circ )\) ; the first concerns finding suitable rings of definition, while the second concerns a description via formal models.
First, we claim that there exists an open and bounded tfp \(\mathcal {O}_K\) -subalgebra \(A_0 \subset A^0\)  such that \(\mathrm {Spec}(\tilde{A}) \rightarrow \mathrm {Spec}(A_0/tA_0)\)  is a universal homeomorphism and such that \(A^\circ \)  is the integral closure of \(A_0\)  in \(A\) . In fact, the second part follows from [1]}. For the first, using Noether normalization, we can choose a surjective map \(T \rightarrow A\) , where \(T = K \langle x_1,...,x_n \rangle \)  is a Tate algebra. By [1]}, the map \(\tilde{T} \rightarrow \tilde{A}\)  is module finite. Setting \(A^{\prime }_0 = \mathrm {im}(T^\circ \rightarrow A^\circ )\) , we learn that \(A_0^{\prime }/(A^{\circ \circ } \cap A_0^{\prime }) \rightarrow \tilde{A}\)  is module finite and injective, with \(A_0^{\prime }\)  being open, bounded, and tfp over \(\mathcal {O}_K\) . Enlarging \(A_0^{\prime }\)  inside \(A^\circ \)  by adding finitely topological generators of \(\tilde{A}\)  over \(A_0^{\prime }\) , we find an open bounded tfp \(\mathcal {O}_K\) -subalgebra \(A_0 \subset A^\circ \)  such that \(A_0/(A^{\circ \circ } \cap A_0) \rightarrow \tilde{A}\)  is bijective. But we also know that \(\sqrt{tA_0} = A^{\circ \circ } \cap A_0\) : any element of the right side is topologically nilpotent and in \(A_0\) , and must thus have a large enough power inside \(tA_0\) . Thus, we have found the subalgebra \(A_0\)  indicated at the start of this paragraph.
Next, we also recall an alternative description of the locally ringed space \((\mathrm {Spa}(A,A^\circ ), \mathcal {O}^+)\) . Consider the category of all proper maps \(f_i:X_i \rightarrow \mathrm {Spec}(A_0)\)  of schemes which are isomorphisms after inverting \(t\) . For each \(f_i\) , let \(X_{i,t=0} \subset X_i\)  be the special fibre (regarded merely as a closed subset). Set \(Z = \lim _i X_{i,t=0}\) , so \(Z\)  is a spectral space. Let \(\pi _i:Z \rightarrow X_i\)  be the structure map, and define the structure sheaf \(\mathcal {O}_Z\)  of \(Z\)  via \(\mathcal {O}_Z := \mathop {\mathrm {colim}}_i \pi _i^{-1} \mathcal {O}_{X_i}\) . Then it is a basic fact that \(\mathrm {Spa}(A,A^\circ ) = Z\)  as topological spaces, and \(\mathcal {O}^+\)  identifies with the \(t\) -adic completion of \(\mathcal {O}_Z\) . For future use, we remark that, by passing to a cofinal subsystem, we may (and do) assume that each \(X_i\)  is \(\mathcal {O}_K\) -flat, i.e., \(\mathcal {O}_{X_i}\)  is \(t\) -torsionfree. This condition implies that the generic fibre \(X_i[1/t]\)  is dense in each \(X_i\) , and thus all the transition maps \(X_i \rightarrow X_j\)  in the system are surjective: their image is a closed set containing a dense open. In particular, \(\mathcal {O}_Z\)  is \(t\) -torsionfree. By spectrality, the maps \(\pi _i:Z \rightarrow X_{i,t=0}\)  are also surjective for all \(i\) . Finally, we also remark that \(\mathcal {O}_Z\)  is integrally closed in \(\mathcal {O}_Z[1/t]\)  by generalities on blowups.
Using the preceding two observations, we prove the equivalences.
\((1) \Rightarrow (2)\) : Fix \(x \in \mathrm {Spa}(A)\)  with \(\mathop {\mathrm {sp}}(x) \in \mathop {\mathrm {Spec}}\tilde{A}\)  being a generic point. Using a suitable Noether normalization, we then learn that \(\tilde{A}_{\mathop {\mathrm {sp}}(x)}\)  is a rank 1 valuation ring with pseudouniformizer \(t\) : this ring is the integral closure of a rank 1 \(t\) -complete valuation ring in a finite extension of its fraction field. Any point \(y \in \mathop {\mathrm {sp}}^{-1}(\mathop {\mathrm {sp}}(x))\)  is represented by an equivalence class of maps \(A_0 \rightarrow V\)  where \(V\)  is a \(t\) -complete \(t\) -torsionfree valuation ring with the property that the closed point of \(\mathrm {Spec}(V)\)  is carried to \(\mathop {\mathrm {sp}}(x)\) . But any such map factors uniquely as \(A_0 \rightarrow \tilde{A}_{\mathop {\mathrm {sp}}(x)} \rightarrow V\) . Thus, taking \(V\)  to be the \(t\) -completion of \(\tilde{A}_{\mathop {\mathrm {sp}}(x)}\)  gives the unique such map up to equivalence, showing that \(y=x\) .
\((2) \Rightarrow (3)\) : Write \(x_i \in X_i\)  for the image of \(x\) , so \(\mathcal {O}^{+}_{\mathrm {Spa}(A,A^+),x}\)  and \(\mathop {\mathrm {colim}}\mathcal {O}_{X_i,x_i}\)  identify after \(t\) -completion. Now if \(x = \mathop {\mathrm {sp}}^{-1}(\mathop {\mathrm {sp}}(x))\) , then \(x_i = f_i^{-1}(\mathop {\mathrm {sp}}(x))\)  is the unique preimage of \(\mathop {\mathrm {sp}}(x)\) . The map \(A_{0,\mathop {\mathrm {sp}}(x)} \rightarrow \mathcal {O}_{X_i,x_i}\)  is then integral (by properness of \(X_i \rightarrow \mathrm {Spec}(A_0)\) , base changed to \(\mathrm {Spec}(A_{0,\mathop {\mathrm {sp}}(x)}\) ) and an isomorphism after inverting \(t\) . Taking a colimit, we learn that \(A_{0,\mathop {\mathrm {sp}}(x)} \rightarrow \mathop {\mathrm {colim}}_i \mathcal {O}_{X_i,x_i}\)  is integral and an isomorphism after inverting \(t\) . But the target is also integrally closed in its \(t\) -localization, so it must coincide with \(A^{\circ }_{\mathop {\mathrm {sp}}(x)}\) ; here we implicitly use that \(\mathrm {Spec}(\tilde{A}) \simeq \mathrm {Spec}(A_0/t)\)  to identify \(\mathop {\mathrm {sp}}(x)\)  with a point of \(\mathrm {Spec}(A^\circ )\) , as well as the fact that \(A^\circ \)  is the integral closure of \(A_0\) . Thus, we have shown that the \(t\) -completion of \(A^{\circ }_{\mathop {\mathrm {sp}}(x)} \rightarrow \mathcal {O}^{+}_{\mathrm {Spa}(A,A^+),x}\)  is an isomorphism. As the \(t\) -completion of the target is \(K_x^+\) , the claim follows.
\((3) \Rightarrow (2)\) : This is clear from the description of points of \(\mathrm {Spa}(A,A^\circ )\)  as equivalence classes of maps \(A^\circ \rightarrow V\)  to \(t\) -complete and \(t\) -torsionfree valuation rings.
\((2) \Rightarrow (1)\) : Assume that \(x\)  is the unique preimage of \(\mathop {\mathrm {sp}}(x)\)  and yet that \(\mathop {\mathrm {sp}}(x) \in \mathrm {Spec}(\tilde{A})\)  is not a generic point; we shall obtain a contradiction. Let \(\overline{y} \in \mathrm {Spec}(\tilde{A})\)  be a generic point of an irreducible component \(Y \subset \mathrm {Spec}(\tilde{A})\)  containing \(\mathop {\mathrm {sp}}(x)\) . Applying the implications \((1) \Rightarrow (3)\)  to a lift \(y \in \mathrm {Spa}(A,A^+)\)  of \(\overline{y}\) , we learn that \(\overline{y}\)  has a unique lift \(y \in \mathrm {Spa}(A,A^+)\) , and that \(K_{y}^+\)  is the \(t\) -completed local ring of \(A^{\circ }_y\) . In particular, the residue field of \(K_{y}^+\)  identifies with function field \(K(Y)\)  of the irreducible component \(Y\) . Choose a valuation subring \(\overline{V} \subset K(Y)\)  that is a \(\tilde{A}\) -algebra and has center \(\mathop {\mathrm {sp}}(x) \in \mathrm {Spec}(\tilde{A})\) ; this valuation has rank \(\ge 1\)  as \(\overline{y} \ne \mathop {\mathrm {sp}}(x)\) . Let \(V \subset K_{y}^+\)  be the preimage of \(\overline{V}\) . Then \(V\)  is a \(t\) -complete valuation ring of rank \(\ge 2\) , is an \(A^\circ \) -algebra, and has center \(\mathop {\mathrm {sp}}(x)\)  on \(\mathrm {Spec}(\tilde{A})\) . The resulting map \(A^\circ \rightarrow V\)  then gives a point \(x^{\prime } \in \mathrm {Spa}(A,A^\circ )\)  with rank \(\ge 2\)  and image \(\mathop {\mathrm {sp}}(x)\)  in \(\mathrm {Spec}(\tilde{A})\) . But \(x\)  is the unique preimage of \(\mathop {\mathrm {sp}}(x)\) , so \(x = x^{\prime }\) . We now obtain a contradiction as \(x\)  had rank 1 by assumption, while \(x^{\prime }\)  has rank \(\ge 2\)  by construction.
\((1) \Longleftrightarrow (4)\) : This is proven in [3]}.
